COUPLED MODEL FOR THE INTERIOR TYPE PERMANENT MAGNET
SYNCHRONOUS MOTORS AT DIFFERENT SPEEDS
M. Pérez-Donsión
Electrical Engineering Department, Vigo University, Campus of Lagoas-Marcosende, 36200 VIGO (Spain)
Phone/Fax: (+34986) 812685 e-mail: donsion@uvigo.es
1. INTRODUCTION
Permanent Magnet Synchronous Motors
(PMSM) are widely applied in industrial and
robotic applications due to their high efficiency,
low inertia and high torque-to-volume ratio.
Concerning with the design one of the greatest
advantages of PMSM is that it can be designed
directly for low speeds without any weakening in
efficiency or power factor. An induction motor
with a mechanical gearbox can often be replaced
with a direct PMSM drive. Both space and cost will
be saved, because the efficiency increases and the
cost of maintenance decreases. A PMSM and a
frequency converter form together a simple and
effective choice in variable speed drives, because
the total efficiency remains high even at lower
speeds and the control of the whole system is very
accurate. Since a low speed motor requires often a
large amount of poles the number of stator slots per
pole and phase is typically low. Thus the stator
magneto motive force contains a lot of large
harmonic components. Especially the fifth and the
seventh stator harmonics are very harmful and tend
to produce torque ripple at a frequency six times
the supply frequency. At the lowest speed this
might be extremely harmful.
The classical d-q model, uncoupled, linear and
with constant parameter, applied to salient pole
synchronous machines may be inadequate for
accurate modelling and characteristics prediction of
permanent magnet synchronous motors of interior
type. It leads to important errors when evaluating
machine performance or calculating the control
circuits.
The lack of excitation control is one of the most
important features of permanent magnet motors, as
a consequence, the internal voltage of the motor
rises proportionally to the rotor speed, and when
the motor is working at constant horsepower mode
its power factor becomes leading.
The behaviour of permanent magnet machines of
the interior type can be rather different than
expected form the conventional two axis theory.
For this reason, it is necessary to establish new
models to take into account the magnetic flux
redistribution phenomena along the rotor iron placed
between the magnets and the air-gap.
On the other side due to the presence of permanent
magnet excitation, the conventional methods of testing
for determination of synchronous machine parameters
can not be applied in the case of permanent magnet
machines, then it is necessary use tests procedures that
differ from the classical methods applicable to wound
field synchronous machines.
In order to observe the cross coupling phenomenon, we
can measure and plot the curves of the interior voltage of
the motor, “Vqi” versus “Id”, for the machine under
study, Fig. 1.
Figure 1.- Graphic representation of Vqi versus Id
The voltage steady state equations will be:
Vqi = α ⋅ Vq − R1 ⋅ Iq = α ⋅ Eo + α ⋅ Xd ⋅ Id
Vdi = α ⋅ Vd − R1 ⋅ Id = α ⋅ Xq ⋅ Iq
(1)
Where “R1” is the stator resistance, “E” is the induced
voltage by the magnets and α is a coefficient for take into
account the operation at different speeds.
If the cross coupling effect didn’t exist and considering
constant excitation all curves Vqi=f(Id) should cross at
the same point for Id=0. However they intersect in
different points. We can see , Fig. 1, that for Id=0 the
distance between two curves Vqi is proportional to Iq,
then we can think it is due to the magnetic coupling
between d-q axis circuits, or in other words, the magnetic
effects on the d-axis flux caused by q-axis current, of
course we can consider the influence on the q-axis flux
motivated by d-axis current. A possible solution for take
into account this effect consist in the addition of a
coupling term between the direct and the quadrature axis,
then the model becomes:
Vqi = α ⋅ Eo + α ⋅ Xd ⋅ Id + α ⋅ Xqd ⋅ Iq
(2)
Vdi = α ⋅ Xq ⋅ Iq + α ⋅ Xdq ⋅ Id
The effect of the term α ⋅ Xdq ⋅ Id depends of the
configuration and dimensions of the PMSM and for
the SIEMOSYN motors, Fig. 2, we have observed
that it is practically negligible.
In this method, the MSIP operate like a generator, at
synchronous speed, over a balanced three phase load.
First we text the machine without load, we take the
measurement of the Eo voltage and establish the
position of the q-axis. After that we apply at
synchronous machine different loads and we obtain the
load angle in each case. In Fig.4 we can see the text
scheme for this method.
SUPLY
VOLTAGE
SUPLY
VOLTAGE
THREE PHASE
LOAD
ELECTRICAL
SIGNAL
ANALYZER
Figure 2.- Rotor configuration of a SIEMOSYN interior type
PMSM
SYNCHRONOUS
MACHINE
DC MOTOR
PMSM
DYNAMIC
SIGNAL
ANALYZER
Then we can consider that the definition equations,
Vqi and Vdi, for a SIEMOSYN PMSM, are:
Figure 4.- Text scheme load-angle method.
Vqi = α ⋅ Eo + α ⋅ Xd ⋅ Id + α ⋅ Xqd ⋅ Iq
Vdi = α ⋅ Xq ⋅ Iq
(3)
and in Fig. 3 we can see the phasor diagram
Figure 3. Phasor diagram for the SIEMOSYN interior type
PMSM
II SYNCHRONOUS REACTANCES
Due to the presence of permanent excitation, the
conventional methods of testing for determination of
synchronous machine parameters can not be applied
in the case of a permanent magnet machine.
Measurement of its electrical parameters requires test
procedures that differ from the classical methods
applicable to wound field synchronous machines.
LOAD-ANGLE METHOD.
Taking into account the classical model and for
different speeds (different frequencies), the phasor
diagram is represented in Fig. 5.
Figure 5. Phasor diagram model for a synchronous generator of
salient poles at different speeds.
And then the equations of the voltages over the d and q
axis, are:
V ⋅ Sin(− δ ) = α ⋅ Xq ⋅ Iq − R1 ⋅ Id
V ⋅ Cos (− δ ) = α ⋅ Eo − α ⋅ Xd ⋅ Id − R1 ⋅ Iq
(4)
For currents:
Id = I 1 ⋅ Sin(φ − δ )
Iq = I 1 ⋅ Cos (φ − δ )
(5)
Replacing the d-q currents, into voltage equations, allows
solution to direct and quadrature axis reactances, for α=1:
Xd = [Eo − V ⋅ Cos (− δ ) − R1 ⋅ I 1 ⋅ Cos (φ − δ )] /
/ I 1 ⋅ Sin(φ − δ )
Xq = [V ⋅ Sin (− δ ) − R1 ⋅ I 1 ⋅ Sin (φ − δ
/ I 1 ⋅ Cos (φ − δ
)
Xd (p.u)
)]/
(6)
Where: α= actual frequency/base frequency,
δ=load angle and Φ=power factor angle.
Using the expressions (6) we can calculate the
reactances taken measurements for obtain the values
of V, I1, P, Cosφ and also the load-angle (δ ) .
Without load this angle is δ 0 , Fig. 6. The load angle
along the successive load test is calculated comparing
the waveforms of the voltage supply and the
reference signal.
Id (p.u)
Figure 7. Graphic representation of Xd versus Id.
The values of the direct axis reactance, Xd, calculated
by the equation 6 are not in agreement with the expected
values of this reactance. We think this is because the daxis flux consist of the combine action of magnets, d-axis
current and q-axis current. The effect of Iq can be
magnetizing or demagnetizing depending of the rotor
geometry and it is not possible to separate by test the
individual contributions of the magnet and the Id current
to the total d-axis flux.
Xd (p.u); Xda (p.u)
Figure 6.- Charts for determination of the reference angle
δ0 .
In Fig. 7 we have represented the results obtained
for the quadrature reactance Xq. Like we can see that
the results are not constant if the Iq current change.
We also have obtained this values by other procedure
(current method) and we can conclude that both
procedures are in a good agreement. This results are
also in concordance with the obtained by other
authors for PMSM of the interior type but with
different geometries. Then we can say that this
phenomena is common for all the interior type
PMSM.
Id (p.u.)
Figure 8. Graphic representation of Xd versus Id.
+ Values of Xd according with the classical model
- Values of Xd take into account the cross coupling
In Fig. 8 we can see Xd values versus Id applied the
classical model and calculated by the following equation
(coupled model):
Xda = [Vd − Eo − R1 ⋅ Iq − Xdq ⋅ Iq ] / Id
Xq (p.u)
(7)
In Fig. 8 we can observe that the values of Xd with cross
coupling are practically constant, which implies that, in
this case, the most of the flux path on the d-axis is
produced by the magnets.
Iq (p.u.)
Figure 7. Graphic representation of Xq versus Iq.
In reference [4] we have developed the Xqd reactance
determination and we have compared, in different cases,
the simulation results using the classical model and the
coupled model with the real measurements and we
concluded that the values calculated using the coupled
model are in better agreement with those obtained by text.
III PMSM BEHAVIOR
Figure 8. Graphic representation of speed versus time during the
started process, obtained by: -.- Applied the model (simulation)
and taken measurements (continuous line).
Now we have developed new texts and simulations
for analyse other cases of the real operation of the
PMSM. Then Fig. 8 show the good concordance
between the curves speed-time obtained by
simulation and by text. In this case we have used an
acceleration ramp of 0 to 50 Hz during 0.45 seconds
take into account a friction and ventilation torque of
0.011 p.u. and without load. It is curious observe the
initial negative interval of the speed which depend on
the initial angle between one of the motor phases and
the direct axis. The effect of the saturation on the qaxis is take into account using the variation of the qreactance with the q-axis current obtained by text.
Figure 9. Graphic representation of speed versus time during a
load sudden increase, obtained by: -.- Applied the model
(simulation) and taken measurements (continuous line).
In Fig. 9 we can observe the incidence that over
the speed has a 0.25 p.u. sudden increase of the load
and in Fig. 10 the influence that produce a sudden
decrease of load, when previously the machine has
obtained the permanent regimen.
The Fig. 11 represent the temporal evolution of the
speed just after has take place a overload Sc, for
different values of the permanent load torque before
Figure 10. Graphic representation of speed versus time during a load
sudden decrease, obtained by: -.- Applied the model (simulation) and
taken measurements (continuous line).
the disturbance. The sudden application of the load
produce an instantaneous decrease of the speed and then
appear an positive asynchronous torque (Fig.14) that
helps to the rotor obtain one time more the synchronism.
This asynchronous torque disappear just in the moment
that the rotor obtain the synchronization. Like one can
observe in Fig. 11 with the same value of the overload,
the maximum slip obtained is lower for the higher level
of the stationary initial load torque. At the same time this
slip is so higher as so higher is the overload value and in
consequence, for the same final load, so higher is the
overload as higher is the maximum slip obtained. At the
same time we can also observe that the time for which the
maximum slip is obtained is practically the same in all
cases.
Figure 11 Graphic representation of speed versus time during a load
sudden increase.
It is interesting take notice in Fig. 11 that, one time that
the motor obtain the synchronization, it can permit the
application of sudden loads higher than it can synchronise
when it start for the same inertia.
Then one of the most important factors that has
influence about the transient behaviour of the PMSM in
front of a sudden increase/decrease of the load is the rotor
inertia. A high value of the rotor inertia produce a large
number of oscillations and if the value of the inertia is
lower the response is more quicker, because the ratio
torque/inertia is higher, but with the maximum slip
more higher, Fig. 12.
T (p.u.)
Figure 15. Graphic representation of the magnets and reluctance
torques versus time during a sudden increase of the load.
T (p.u.)
Figure 12. Graphic representation of speed versus time for
different inertia torque (M) with Tl=0 and Sl=1.
Tsc (p.u.)
T (s)
Figure 16. Graphic representation of the magnets and reluctance
torques versus time during a sudden decrease of the load.
Figure 13. Graphic representation of the torque of the squirrel
cage versus time for a sudden decrease of the load.
In Fig. 13 we have represented the squirrel cage
torque when take place a sudden decrease of the load
and in Fig. 14 when the load increase. In both cases
for the same values of the load torque (Tl) and overload (Sl).
In Fig. 15 we have represented the torque of the magnets
and reluctance when take place a sudden increase of the
load and in Fig. 16 when the load decrease. In both cases
for the same values of the load torque (Tl) and over-load
(Sl). Logically the synchronous torques of permanent
magnets and reluctance permit maintain the rotor in
synchronism.
Tsc (p.u.)
Figure 14. Graphic representation of the torque of the squirrel
cage versus time for a sudden increase of the load.
Figure 17. Graphic representation of speed versus time for different
values of the equivalent current of the magnets with Sl=1 and Tl=0.
The permanent magnets influence about the transient
behaviour of the PMSM is very important. As higher are
the equivalent currents of the magnets as lower are
the slips of the transient response. In Fig. 17 we can
observe that if the current of the magnet decrease
below a certain value the motor is not capable of take
up the overload and the motor lost the synchronism.
The optimum value of this current depends, amount
other factors, of the magnets braking torque at the
synchronous speed proximity. If this value is
overcome they will appear higher oscillations during
the transient operation.
The rotor geometry and in consequence the
relationship between the d-axis and q-axis reactances,
also modify the PMSM behaviour, as in the same
way that for the equivalent current we must obtain an
optimum value for the relation Xd/Xq and also it is
important to obtain the most appropriate squirrel cage
resistance value.
Really the number of variables that have influence
about the starting and synchronization processes of a
PMSM, take into account the motor and also the
load, is very higher and then it is very difficult know
in advance a set of necessary conditions for the
correct synchronization of the PMSM. Then for
develop an analyse of this type it is necessary take
into account the parametric variation of the main
magnitudes that have influence about the
synchronization process.
In this particular case we have analyzed this
process in term of his synchronization energy,
specially we have considered the property “capacity
of synchronization” of the motor, that we can defined
it like a set of critical combinations of inertia and
load torque in which the PMSM is capable to obtain
the synchronization.
For obtain the synchronization energy we use a set
of simple expressions that permit determine this
magnitude in the last stage of the synchronous
operation of the motor just when the machine
describe a limit circle.
The dynamic equation expressed in the torque-slip
plane is, (8):
−
1
ds
J .w02 ⋅ s
= Ts (δ ) + Ta (s ) − Tc ( s ) (8)
p
dδ
Where: J is the combination inertia of the motor
and the load, Ts is the sum of all the synchronization
torques, Ta include all the asynchronous average
torques and Tc is the sum of the load, slip and
ventilation torques.
The equation (8) describe the critical trajectories of
the polar slips on the load angle-slip plane.
IV CONCLUSIONS
We have developed along this paper a coupled model for
accurate representation of the characteristics of
permanent magnet synchronous motors and we have
proposed the determination of the direct axis reactance,
“Xd”, and the quadrature axis reactance, “Xq”, by
calculus and texts with the permanent magnet
synchronous machine under generator duty.
We also have analysed the starting and synchronization
processes of the PMSM and the influence that on
transient behaviour of the motor produce different values
of the main motor parameters.
V. REFERENCES.
[1] Salo J., Heikkilä T., Pyrhönen, Haring T “New LowSpeed High-Torque Permanent Magnet Synchronous
Machine With Buried Magnets” International
Conference on Electrical Machines (ICEM 2000), pp.
1246-1250, Espoo, Finland.
[2] Parasiliti F., Poffet P., “A model for saturation
effects in high field permanent magnet synchronous
motors”, IEEE Trans. On Energy Conversión, Vol.4,
Nº.3, pp.487-494, Sep. 1989.
[3] Donsion, M.P., Ferro M.F. “Motores sincronos de
imanes permanentes”, Research book published by the
University of Santiago de Compostela, Spain.
[4] Donsión M.P., Manzanedo J.F., Iglesias C. “Coupled
model of the interior type permanent magnet synchronous
motor. Application to a Siemosyn motor”, pp. 144-147,
International Conference on Electrical Machines
(ICEM’94), París, France.
[5] Ferro M.F., Donsion, M.P. “Torques Analysis in
Permanent Magnet Synchronous Motors”, IASTED
Power High Tech’89, pp. 271-275, Valencia, Spain.
[6] Ferro M.F., Donsión M.P.“Transient behavior of
permanent magnet synchronous motors under sudden
change in load”, IASTED Ninth International
Symposium, Modelling, Identification and Control, pp.
406-410 Innsbruck, Austria.
[7] Ferro M.F., Donsión, M.P..“Specific Characteristics
of the interior type permanent magnet synchronous
motors. Aplication so a Siemosyn 1FU3134". International AEGEAN Conference on Electrical Machines and
Power Electronics,Vol. 2, pp.378-382, Turkey.